Row-Column Response Surface Designs

Factorial experiments are widely used in industry
to investigate the effects of process factors on quality response
variables. Many food processes, for example, are not only
subject to variation between days, but also between different
times of the day. Removing this variation using blocking factors
leads to rowcolumn designs. In this paper, an algorithm is
described for constructing factorial row-column designs when
the factors are quantitative, and the data are to be analysed
by fitting a polynomial model. The row-column designs are
constructed using an iterative interchange search, where interchanges
that result in an improvement in the weighted mean of the
efficiency factors corresponding to the parameters of interest
are accepted. Some examples illustrating the performance of
the algorithm are given.

RESPONSE surface methodology (RSM) is a set of
techniques for empirically studying processes
by designing experiments, modelling the resulting
data, and interpreting the .tted models. At least
initially, low order polynomial models are usually fitted,
often the second order model,

where y is the response variable, x1, ..., xq are the
levels of the continuous factors being varied, coded
to be between -1 and 1, and is the
error term. For a full discussion of RSM, see the
books by Box and Draper (1987), Myers and Montgomery
(1995), and Khuri and Cornell (1996). For
more recent developments, see Myers (1999).

Traditionally, response surface (RS) designs have
been described as if the treatments are completely
randomised to the experimental units, but recently
there has been interest in arranging these designs in
other unit structures. Atkinson and Donev (1989),
Cook and Nachtsheim (1989), and Trinca and
Gilmour (2000) gave algorithms for arranging RS designs
in blocks. Trinca and Gilmour (2001) gave
an algorithm for arranging RS designs in general
nested unit structures, such as split plots. Edmondson
(1991) showed how four-level RS designs, based
on two-level pseudo-factors, could be arranged in various
unit structures using confounding of the appropriate
e.ects in the pseudo-factors. Until now, there
has been no discussion of how to arrange RS designs
in general row-column or other crossed unit structures.
This is the subject of the current paper. In
the second section, we discuss food processing and
other applications. Next, our criterion and algorithm
are described. The algorithm is applied in the fourth
section to three examples, data analysis is discussed
in the fifth section, and in the sixth section some
practical issues and extensions are discussed.